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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sufficient conditions for smoothing codimension one foliations


Author: Christopher Ennis
Journal: Trans. Amer. Math. Soc. 276 (1983), 311-322
MSC: Primary 57R30; Secondary 57R10, 58F18
MathSciNet review: 684511
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Abstract: Let $ M$ be a compact $ {C^\infty}$ manifold. Let $ X$ be a $ {C^0}$ nonsingular vector field on $ M$, having unique integral curves $ (p,t)$ through $ p \in M$. For $ f: M \to {\mathbf{R}}$ continuous, call $ \left. Xf(p) = df(p,t)/dt\right\vert _{t = 0}$ whenever defined. Similarly, call $ {X^k}f(p)=X(X^{k-1}f)(p)$.

For $ 0 \leqslant r < k$, a $ {C^r}$ foliation $ \mathcal{F}$ of $ M$ is said to be $ {C^k}$ smoothable if there exist a $ {C^k}$ foliation $ \mathcal{G}$, which $ {C^r}$ approximates $ \mathcal{F}$, and a homeomorphism $ h:M \to M$ such that $ h$ takes leaves of $ \mathcal{F}$ onto leaves of $ \mathcal{G}$.

Definition. A transversely oriented Lyapunov foliation is a pair $ (\mathcal{F},X)$ consisting of a $ {C^0}$ codimension one foliation $ \mathcal{F}$ of $ M$ and a $ {C^0}$ nonsingular, uniquely integrable vector field $ X$ on $ M$, such that there is a covering of $ M$ by neighborhoods $ \{{W_i}\} $, $ 0 \leqslant i \leqslant N$, on which $ \mathcal{F}$ is described as level sets of continuous functions $ {f_i}:{W_i} \to {\mathbf{R}}$ for which $ X{f_i}(p)$ is continuous and strictly positive.

We prove the following theorems.

Theorem 1. Every $ {C^0}$ transversely oriented Lyapunov foliation $ (\mathcal{F},X)$ is $ {C^1}$ smoothable to a $ {C^1}$ transversely oriented Lyapunov foliation $ (\mathcal{G},X)$.

Theorem 2. If $ (\mathcal{F},X)$ is a $ {C^0}$ transversely oriented Lyapunov foliation, with $ X \in {C^{k - 1}}$ and $ {X^j}{f_i}(p)$ continuous for $ 1 \leqslant j \leqslant k$ and $ 0 \leqslant i \leqslant N$, then $ (\mathcal{F},X)$ is $ {C^k}$ smoothable to a $ {C^k}$ transversely oriented Lyapunov foliation $ (\mathcal{G},X)$.

The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the $ {C^k}$ version of Theorem 1.

Theorem 3. If $ (\mathcal{F},X)$ is a $ {C^{k - 1}}\;(k \geqslant 2)$ transversely oriented Lyapunov foliation, with $ X \in {C^{k - 1}}$ and $ {X^k}{f_i}(p)$ is continuous, then $ (\mathcal{F},X)$ is $ {C^k}$ smoothable to a $ {C^k}$ transversely oriented Lyapunov foliation $ (\mathcal{G},X)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0684511-4
PII: S 0002-9947(1983)0684511-4
Article copyright: © Copyright 1983 American Mathematical Society